Tetrahedron in a unit sphere

Hi everyone, i'm in need of help with a certain aspect of my project, hopefully someone has some suggestions:

Im doing a mesh morphing program for two arbitrary genus0 meshes, i'm using the method of Marc Alexa for implementation.

However i am stuck on a certain part which says:

'Choose a random regular tetrahedron with vertices on the unit sphere'.

Ive been searching for a few days now to find out how to construct a regular tetrahedron which fits exactly (more or less) inside a unit sphere.

From what i have read, i need to have a 'circumscribed' tetrahedron', which has the sphere going outside through all the vertices in the tetrahedron.

But i am having great difficulty programming this as i cannot work out how to determine how big the tetrahedron should be so that the vertices are on the surface of the sphere.

Any help or suggestions would be greatly appreciated.

Many Thanks

Dan